Schrage announced his departure from Facebook in June.
"He agreed to take on the job after months of wooing by Mark Zuckerberg, Facebook's chief executive, who told Mr Clegg he would have a leading role in shaping the company's strategy", reported the Financial Times.
In his new role at Facebook, Mr Clegg will report to chief operating officer Sheryl Sandberg.
Anna Burns says Booker Prize-winning novel was not intentionally political
It was originally open to British, Irish and Commonwealth writers but Americans have been eligible since 2014. Booker's chair of judges, the philosopher Kwame Anthony Appiah, said that the novel is "incredibly original".
"It's time we pause for breath before everyone charges off in a stampede of condemnation of tax-dodging-fake-news-extremism-promoting-data-controlling tech firms", he wrote. "I hope I will be able to play a role in helping to navigate that journey".
The appointment comes amid intense scrutiny of the company and the threat of government regulation following allegations of election meddling and the Cambridge Analytica data scandal. "I find the messianic Californian new-worldy-touchy-feely culture of Facebook a little grating".
Facebook has faced a barrage of criticism from users and lawmakers after it said in 2017 that Russian agents used its platform to spread disinformation before and after the 2016 US presidential election, an accusation Moscow denies.
Last week, the company said hackers had stolen personal details of 29 million Facebook users in September. He is set to remain present at Facebook as an adviser to Clegg and the rest of the leadership team over the coming years.